1. Section 3.2 Mean Value Theorem Math 1231: Single-Variable Calculus

2. Rolle’s Theorem

4. Examples Prove that the equation x 3 + x - 1 =0 has exactly one real root.

5. When does Rolle’s theorem fail? When f is discontinuous on the closed interval [a, b], or f is non- differentiable at an interior point of the interval (a, b), the conclusion of Rolle’s theorem may not hold.

6. Mean Value Theorem

7. There is at least one point P(c, f(c)) on the graph where the slope of the tangent line is the same as the slope of the secant line AB.

8. Examples

9. Consequence from MVT